Average Error: 14.8 → 0.4
Time: 43.3s
Precision: 64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r \cdot \sin b}{\cos a \cdot \cos b - \sqrt[3]{\left(\sin a \cdot \sin b\right) \cdot \left(\left(\sin a \cdot \sin b\right) \cdot \left(\sin a \cdot \sin b\right)\right)}}\]
r \cdot \frac{\sin b}{\cos \left(a + b\right)}
↓
\frac{r \cdot \sin b}{\cos a \cdot \cos b - \sqrt[3]{\left(\sin a \cdot \sin b\right) \cdot \left(\left(\sin a \cdot \sin b\right) \cdot \left(\sin a \cdot \sin b\right)\right)}}
double f(double r, double a, double b) {
        double r1006972 = r;
        double r1006973 = b;
        double r1006974 = sin(r1006973);
        double r1006975 = a;
        double r1006976 = r1006975 + r1006973;
        double r1006977 = cos(r1006976);
        double r1006978 = r1006974 / r1006977;
        double r1006979 = r1006972 * r1006978;
        return r1006979;
}
↓
double f(double r, double a, double b) {
        double r1006980 = r;
        double r1006981 = b;
        double r1006982 = sin(r1006981);
        double r1006983 = r1006980 * r1006982;
        double r1006984 = a;
        double r1006985 = cos(r1006984);
        double r1006986 = cos(r1006981);
        double r1006987 = r1006985 * r1006986;
        double r1006988 = sin(r1006984);
        double r1006989 = r1006988 * r1006982;
        double r1006990 = r1006989 * r1006989;
        double r1006991 = r1006989 * r1006990;
        double r1006992 = cbrt(r1006991);
        double r1006993 = r1006987 - r1006992;
        double r1006994 = r1006983 / r1006993;
        return r1006994;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 14.8

    \[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
  2. Using strategy rm

  3. Applied cos-sum0.3

    \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos a \cdot \cos b - \sin a \cdot \sin b}}\]

  4. Taylor expanded around inf 0.3

    \[\leadsto \color{blue}{\frac{\sin b \cdot r}{\cos a \cdot \cos b - \sin b \cdot \sin a}}\]
  5. Using strategy rm

  6. Applied add-cbrt-cube0.4

    \[\leadsto \frac{\sin b \cdot r}{\cos a \cdot \cos b - \sin b \cdot \color{blue}{\sqrt[3]{\left(\sin a \cdot \sin a\right) \cdot \sin a}}}\]

  7. Applied add-cbrt-cube0.4

    \[\leadsto \frac{\sin b \cdot r}{\cos a \cdot \cos b - \color{blue}{\sqrt[3]{\left(\sin b \cdot \sin b\right) \cdot \sin b}} \cdot \sqrt[3]{\left(\sin a \cdot \sin a\right) \cdot \sin a}}\]

  8. Applied cbrt-unprod0.4

    \[\leadsto \frac{\sin b \cdot r}{\cos a \cdot \cos b - \color{blue}{\sqrt[3]{\left(\left(\sin b \cdot \sin b\right) \cdot \sin b\right) \cdot \left(\left(\sin a \cdot \sin a\right) \cdot \sin a\right)}}}\]

  9. Simplified0.4

    \[\leadsto \frac{\sin b \cdot r}{\cos a \cdot \cos b - \sqrt[3]{\color{blue}{\left(\left(\sin a \cdot \sin b\right) \cdot \left(\sin a \cdot \sin b\right)\right) \cdot \left(\sin a \cdot \sin b\right)}}}\]

  10. Final simplification0.4

    \[\leadsto \frac{r \cdot \sin b}{\cos a \cdot \cos b - \sqrt[3]{\left(\sin a \cdot \sin b\right) \cdot \left(\left(\sin a \cdot \sin b\right) \cdot \left(\sin a \cdot \sin b\right)\right)}}\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  (* r (/ (sin b) (cos (+ a b)))))